Heat oil in a large pot (with a lid) over medium heat. When hot, add the onion, and celery. Cook, stirring often, until the vegetables are just softened, for about 5 minutes. Add the garlic, sausage, and thyme, and cook 1 minute more.
Add the stock and bay leaves, and bring mixture to a simmer over high heat. Stir in the lentils, then reduce heat, cover, and cook at a gentle simmer until tender, for about 60 minutes.
Remove and discard the bay leaves. Take off lid and simmer another 15 minutes to thicken.
Ladle the soup into soup bowls. Sprinkle with parsley.
Soup can be prepared 1 day ahead; cool, cover, and refrigerate. Reheat over medium heat.